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Full text of "Treatise On Analysis Vol-Ii"

7   ALGEBRAIC PROPERTIES OF CONVOLUTION       275

Let St = Supp(jUj), and suppose that St is compact except possibly for one
index j. Let/6 3CC(G) and let K be the compact support of/. It is enough to
show that the set of points (xl9..., xn) e Gn which belong to the support

n

Y[ S,- of fa   *   jun (13.21.18) and are such that xvx2 ' *  xn e K, is relatively

i=l

compact in Gn. Now, the conditions xt e S for all /, and XiX2 - xneK,
imply that

v-   <= Q ~ *    ...Q""lkrC~l...C~l

X7 G j- 1         ^1    ^^n            ^j+ 1 J

and this set is compact (12.10.5). Since by hypothesis St is compact whenever
/ ^67, our assertion is proved (3.20.16).

The result of (14.6.2) or (14.6.3) shows that two measures can be convolv-
able without either of them having a compact support. Later (14.10.7) we
shall see examples of unbounded measures which are convolvable.

7. ALGEBRAIC PROPERTIES OF CONVOLUTION

(14.7.1)    Let A, /z, v be three measures on G, and suppose that the pairs
(A, ju) and (A, v) are convolvable. Then so is the pair (A, ju + v) and we have

(14.7.1.1)                           A * (fi + v) = A * /i + A * v.

For by virtue of the relation \fi + v\ g |ju| + |v| we may restrict ourselves
to the case where A, ju and v are positive, and in this case the result follows
immediately from (13.16.1).

Similarly, if (A, v) and (//, v) are convolvable, then so is (A + ju, v) and we
have

(14.7.1.2)                          (A + //)*v = A*v + /*v.

Also it is clear that if the pair (A, $ is convolvable, then so is (#A, fe/z) for
all scalars a and b, and

(14.7.1.3)                           (aX) * (bti=(ab)l * p.

(14.7.2)    Let A, ju, v be three measures ^0 on G.

(i)   J/if/ie sequence (A, /*, v) & convolvable, then so are the sequences (A, /*),
| * |/4 v), Ox, v), (A, |M| * |v|), of/iJ we

(14.7.2.1)                  A * ji * v = (A * /x) * v = A * (p * v).subset A of Nc is relatively compact in Nc if and only if it satisfies